92 



M. Lorenz on the Theory of Light, 



From the equations thus obtained for %, f]y £ the general 

 equation s, applicable to the case in which the components are 

 also dependent on y, can be deduced. They must of necessity 

 assume the following form : — 



dx a> 



1 d*£ 



2 dt 2 ' 



A 2 V- 



de_ i d* 



dy 



dd 



dz 



V 





CO' 



dt 23 



d 2 l 

 dt 23 



(A) 



in which 



and 



*=& + 



! ^ 2 



<fe 2 ' dy* 



+ dz* 



0-^ + ^1 + ^. 

 dx dy dz 



These equations bring us back again to the equations (30), (31), 

 and (34), if the components are independent of y. 



We can further conclude from the symmetrical form of the 

 equations (A), which is unaltered by changing the axes of coor- 

 dinates, that they would still hold good if co } which has been 

 hitherto considered a function of x only, were any function what- 

 ever of x, y, and z. Accordingly they express the general laws of 

 the motion of light in any non -absorbent heterogeneous medium. 



The principles which must serve for calculating the diffraction, 

 reflexion, and refraction of light, or the conditions which must 

 obtain when light passes from one medium into another, can now 

 be easily deduced from these general equations. Let the plane 

 of coordinates yz be the limiting plane for which the conditions 

 are investigated. The equations will then be multiplied by dx 

 and by dxdx, and simply and doubly integrated from x=0 to 

 x = e, where e is a very small quantity. Now, however, all the 

 integrals will disappear except those whose elements become infi- 

 nite, and this will be the case with those only which contain 

 differential coefficients in relation to x. The second equation (A) 



thus gives 



.. - ■ ■-" : . --. 



(35) 



|"^_ Cp=0 and [^]" =t = 0. 

 Similarly, the third equation (A) gives 



. , r^_fl~ 6 = oandur;=o„ 



Ldx dzJ~ =0 *=o 



(36) 



